a. The OLS regression of Y on all variables in the data set is given below. This was done using EViews. The R2 = 0.434 and the F-statistic for the significance of all slopes is equal to 3.169. This is distributed as F(15,62) under the null hypothesis. This has a p-value of 0.0007. Therefore, we reject Ho and we conclude that this is a significant regression. As explained in Sect. 13.6, using BRMR this also rejects the insignificance of all slopes in the logit specification.

Unrestricted Least Squares

LS // Dependent Variable is Y

Sample: 1 78

Included observations: 78

Variable

Coefficient

Std. Error

t-Statistic

Prob.

C

1.272832

1.411806

0.901563

0.3708

BA

0.000398

0.007307

0.054431

0.9568

BS

0.017084

0.020365

0.838887

0.4048

NW

-0.036932

0.025320

-1.458609

0.1497

FI

-0.221726

0.092813

-2.388949

0.0200

PTS

0.178963

0.091050

1.965544

0.0538

MAT

0.214264

0.202497

1.058108

0.2941

MOB

0.020963

0.009194

2.279984

0.0261

MC

0.189973

0.150816

1.259635

0.2125

FTB

-0.013857

0.136127

-0.101797

0.9192

SE

0.188284

0.360196

0.522728

0.6030

YLD

0.656227

0.366117

1.792399

0.0779

MARG

0.129127

0.054840

2.354621

0.0217

CB

0.172202

0.137827

1.249403

0.2162

STL

-0.001599

0.005994

-0.266823

0.7905

LA

-0.001761

0.007801

-0.225725

0.8222

R-squared

0.433996

Mean dependent var

0.589744

Adjusted R-squared

0.297059

S. D. dependent var

0.495064

S. E. of regression

0.415069

Akaike info criter

-1.577938

Sum squared resid

10.68152

Schwarz criterion

-1.094510

Log likelihood

-33.13764

F-statistic

3.169321

Durbin-Watson stat

0.905968

Prob(F-statistic)

0.000702

Plot of Y and YHAT

b. The URSS from part (a) is 10.6815 while the RRSS by including only the cost variables is 14.0180 as shown in the enclosed output from EViews. The Chow-F statistic for insignificance of 10 personal characteristics variables is

F= (14.0180 – 10.6815)/10

10.6815/62 ‘

which is distributed as F(10,62) under the null hypothesis. This has a 5% critical value of 1.99. Hence, we cannot reject Ho. The principal agent theory suggests that personal characteristics are important in making this mortgage choice. Briefly, this theory suggests that information is asym­metric and the borrower knows things about himself or herself that the lending institution does not. Not rejecting Ho does not provide support for the principal agent theory.

TESTING THE EFFICIENT MARKET HYPOTHESIS WITH THE LINEAR PROBABILITY MODEL

Restricted Least Squares

LS // Dependent Variable is Y

Sample: 1 78

Included observations: 78

Variable

Coefficient

Std. Error

t-Statistic

Prob.

FI

-0.237228

0.078592

-3.018479

0.0035

MARG

0.127029

0.051496

2.466784

0.0160

YLD

0.889908

0.332037

2.680151

0.0091

PTS

0.054879

0.072165

0.760465

0.4495

MAT

0.069466

0.196727

0.353108

0.7250

C

1.856435

1.289797

1.439324

0.1544

R-squared

0.257199

Mean dependent var

0.589744

Adjusted R-squared

0.205616

S. D. dependent var

0.495064

S. E. of regression

0.441242

Akaike info criter

-1.562522

Sum squared resid

14.01798

Schwarz criterion

-1.381236

Log likelihood

-43.73886

F-statistic

4.986087

Durbin-Watson stat

0.509361

Prob(F-statistic)

0.000562

c. The logit specification output using EViews is given below. The unre­stricted log-likelihood is equal to —30.8963. The restricted specification output is also given showing a restricted log-likelihood of —41.4729. Therefore, the LR test statistic is given by LR = 2(41.4729 — 30.8963/ = 21.1532 which is distributed as x20 under the null hypothesis. This is sig­nificant given that the 5% critical value of x20 is 18.31. This means that the logit specification does not reject the principal agent theory as personal characteristics are not jointly insignificant.

d. Similarly, the probit specification output using EViews is given below. The unrestricted log-likelihood is equal to —30.7294. The restricted log – likelihood is —41.7649. Therefore, the LR test statistic is given by LR = 2(41.7649 — 30.7294/ = 22.0710 which is distributed as x?0 under the null hypothesis. This is significant given that the 5% critical value of x20 is 18.31. This means that the probit specification does not reject the principal agent theory as personal characteristics are not jointly insignificant.

13.13 Problem Drinking and Employment. The following Stata output replicates the OLS results given in Table 5 of Mullahy and Sindelar (1996, p. 428) for males. The first regression is for employment, given in column 1 of Table 5 of the paper, and the second regression is for unemployment, given in column 3 of Table 5 of the paper. Robust standard errors are reported.

The following Stata output replicates the OLS results given in Table 6 of Mullahy and Sindelar (1996, p. 429) for females. The first regression is for employment, given in column 1 of Table 6 of the paper, and the second regres­sion is for unemployment, given in column 3 of Table 6 of the paper. Robust standard errors are reported.

The estimates reveal that having children of the same sex has a significant and positive effect on the probability of having an additional child. The marginal effects are given by dprobit in Stata. dprobit f dsex ags26l educ_2 educ_3 age drace inc

Probit regression, reporting marginal effects

Number of obs

= 5768

LR chi2 (7)

= 964.31

Prob > chi2

= 0.0000

Log likelihood = —1561.1312

Pseudo R2

= 0.2360

f

dF/dx

Std. Err.

z

P>|z|

x-bar

[95% C. I.]

dsex*

.0302835

.0069532

5.40

0.000

.256415

.016655

.043912

ags26l*

-.1618148

.0066629

-13.22

0.000

.377601

-.174874

-.148756

educ_2*

.0022157

.0090239

0.24

0.808

.717753

-.015471

.019902

educ_3*

.0288636

.0140083

2.45

0.014

.223994

.001408

.056319

age

-.0065031

.0007644

-16.65

0.000

32.8024

-.008001

-.005005

drace*

-.0077119

.0055649

-1.45

0.146

.773232

-.018619

.003195

inc

.0002542

.000241

1.06

0.289

12.8582

-.000218

.000727

obs. P

.1137309

pred. P

.0367557

(at x-bar)

(*) dF/dx is for discrete change of dummy variable from 0 to 1 z and P> |z| correspond to the test of the underlying coefficient being 0

If we replace same sex by its components: same sex female and same sex male variables, the results do not change indicating that having both boys or girls does not matter, see Carrasco (2001,p.391) Table 4, column 2.

d. The 2sls estimates in Table 5, column 5, of Carrasco (2001, p. 392) using as instruments the same sex variables and their interactions with ags26l is given below, along with the over-identification test and the first stage diagnostics:

z and P> |z| correspond to the test of the underlying coefficient being 0

. estat classification

Probit model for inlf

True-

Classified

D

Total

+

348

120

468

–

80

205

285

Total

428

325

753

Classified + if predicted Pr(D) >= .5 True D defined as inlf!= 0

Sensitivity

Specificity

Positive predictive value Negative predictive value

Pr(+| D) Pr(-| ~D) Pr(D| +) Pr(-D| -)

81.31%

63.08%

74.36%

71.93%

False + rate for true —D

Pr(+| ~D)

36.92%

False – rate for true D

Pr(-| D)

18.69%

False + rate for classified +

Pr(-D| +)

25.64%

False – rate for classified –

Pr(D| -)

28.07%

Correctly classified

73.44%

d. Wooldridge (2009, Chapter 17) recommends one obtain the estimates of (fi/a2) from a probit using an indicator of labor force participation. Then comparing those with the Tobit estimates generated by dividing fi by a2. If these estimates are different or have different signs, then the Tobit esti­mation may not be appropriate. Part (c) gave such probit estimates. For (kidslt6) this was estimated at —0.868. From part (b) the tobit estimation gave a fi estimate for (kidslt6) of —894 and an estimate of a2 of 1122. The resulting estimate of (fi/a2) is —0.797. These have the same sign but with different magnitudes.